Understanding Moments

statisticsprobabilitymomentsskewnesskurtosis

Abstraction: Deep explanation of statistical moments — mean, variance, skewness, kurtosis — and why they are called moments

Key points:

"Moment" derives from Latin "movimentum" (motion); the concept mirrors the physics notion of moment of inertia — how mass (probability) is distributed around an axis
Zeroth moment = total probability mass (always 1); 1st moment = mean (location); 2nd central moment = variance (scale/spread)
3rd standardized moment = skewness: measures relative tail size (sign indicates which tail; ~90% of skewness calculation comes from data beyond 1 std dev)
4th standardized moment = kurtosis: measures combined absolute tailedness, NOT peakedness; kurtosis of any normal is 3; excess kurtosis = kurtosis - 3
Higher moments (5th+) simply recapitulate skewness and kurtosis patterns; by convention only the first four named moments are used
Moment-generating function (MGF) uniquely specifies a distribution via Taylor expansion equal to an infinite sum of weighted raw moments

Connections: Gregory Gundersen · Probability Distributions · Statistical Moments · Moment Generating Function